{"title":"具有完全完备性的3查询非自适应PCP","authors":"Subhash Khot, Rishi Saket","doi":"10.1109/CCC.2006.5","DOIUrl":null,"url":null,"abstract":"We study a very basic open problem regarding the PCP characterization of NP, namely, the power of PCPs with 3 non-adaptive queries and perfect completeness. The lowest soundness known till now for such a PCP is 6/8 + epsi given by a construction of Hastad (1997). However, Zwick (1998) shows that a 3-query non-adaptive PCP with perfect completeness cannot achieve soundness below 5/8. In this paper, we construct a 3-query non-adaptive PCP with perfect completeness and soundness 20/27 + epsi, which improves upon the previous best soundness of 6/8 + epsi. A standard reduction from PCPs to constraint satisfaction problems (CSPs) implies that it is NP-hard to tell if a Boolean CSP on 3-variables has a satisfying assignment or no assignment satisfies more than 20/27 + epsi fraction of the constraints. Our construction uses \"biased long codes\" introduced by Dinur and Safra (2002). We develop new 3-query tests to check consistency between such codes. These tests are analyzed by extending Hastad's Fourier methods (1997) to the biased case","PeriodicalId":325664,"journal":{"name":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2006-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"24","resultStr":"{\"title\":\"A 3-query non-adaptive PCP with perfect completeness\",\"authors\":\"Subhash Khot, Rishi Saket\",\"doi\":\"10.1109/CCC.2006.5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study a very basic open problem regarding the PCP characterization of NP, namely, the power of PCPs with 3 non-adaptive queries and perfect completeness. The lowest soundness known till now for such a PCP is 6/8 + epsi given by a construction of Hastad (1997). However, Zwick (1998) shows that a 3-query non-adaptive PCP with perfect completeness cannot achieve soundness below 5/8. In this paper, we construct a 3-query non-adaptive PCP with perfect completeness and soundness 20/27 + epsi, which improves upon the previous best soundness of 6/8 + epsi. A standard reduction from PCPs to constraint satisfaction problems (CSPs) implies that it is NP-hard to tell if a Boolean CSP on 3-variables has a satisfying assignment or no assignment satisfies more than 20/27 + epsi fraction of the constraints. Our construction uses \\\"biased long codes\\\" introduced by Dinur and Safra (2002). We develop new 3-query tests to check consistency between such codes. These tests are analyzed by extending Hastad's Fourier methods (1997) to the biased case\",\"PeriodicalId\":325664,\"journal\":{\"name\":\"21st Annual IEEE Conference on Computational Complexity (CCC'06)\",\"volume\":\"1 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2006-07-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"24\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"21st Annual IEEE Conference on Computational Complexity (CCC'06)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/CCC.2006.5\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"21st Annual IEEE Conference on Computational Complexity (CCC'06)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/CCC.2006.5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A 3-query non-adaptive PCP with perfect completeness
We study a very basic open problem regarding the PCP characterization of NP, namely, the power of PCPs with 3 non-adaptive queries and perfect completeness. The lowest soundness known till now for such a PCP is 6/8 + epsi given by a construction of Hastad (1997). However, Zwick (1998) shows that a 3-query non-adaptive PCP with perfect completeness cannot achieve soundness below 5/8. In this paper, we construct a 3-query non-adaptive PCP with perfect completeness and soundness 20/27 + epsi, which improves upon the previous best soundness of 6/8 + epsi. A standard reduction from PCPs to constraint satisfaction problems (CSPs) implies that it is NP-hard to tell if a Boolean CSP on 3-variables has a satisfying assignment or no assignment satisfies more than 20/27 + epsi fraction of the constraints. Our construction uses "biased long codes" introduced by Dinur and Safra (2002). We develop new 3-query tests to check consistency between such codes. These tests are analyzed by extending Hastad's Fourier methods (1997) to the biased case
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